Graduate Texts in Mathematics 198 Editorial Board S Axler EW Gehring K.A Ribet Springer Science+Business Media, LLC Graduate Texts in Mathematics 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 TAKEUTIIZARING Introduction to Axiomatic Set Theory 2nd ed OXTOBY Measure and Category 2nd ed SCHAEFER Topological Vector Spaces 2nd ed HILTON/STAMMBACH A Course in Homological Algebra 2nd ed MAc LANE Categories for the Working Mathematician 2nd ed HUGHESIPIPER Projective Planes SERRE A Course in Arithmetic TAKEUTIIZARING Axiomatic Set Theory HUMPHREYS Introductionto Lie Algebras and Representation Theory COHEN A Course in Simple Homotopy Theory CONWAY Functions ofOne Complex Variable I 2nd ed BEALS Advanced Mathematical Analysis ANDERSONIFULLER Rings and Categories ofModules 2nd ed GOLUBITSKy/GUILLEMIN Stable Mappings and Their Singularities BERBERIAN Lectures in Functional Analysis and Operator Theory WINTER The Structure ofFields ROSENBLATT Random Processes 2nd ed HALMOS Measure Theory HALMOS A Hilbert Space Problem Book 2nd ed HUSEMOLLER Fibre Bundles 3rd ed HUMPHREYS Linear Algebraic Groups BARNESIMACK An Algebraic Introduction to Mathematical Logic GREUB Linear Algebra 4th ed HOLMES Geometrie Functional Analysis and Its Applications HEWITT/STROMBERG Real and Abstract Analysis MANEs Algebraic Theories KELLEY General Topology ZARlSKilSAMUEL Commutative Algebra VoLL ZARlSKilSAMUEL CommutativeAlgebra Vol.ll JACOBSON Lectures in Abstract Algebra I Basic Concepts JACOBSON Lectures in Abstract Algebra II Linear Algebra JACOBSON Lectures in Abstract Algebra III Theory ofFields and Galois Theory HIRSCH Differential Topology SPITZER Principles ofRandom Walk 2nd ed ALEXANDERIWERMER Several Complex Variables and Banach Algebras 3rd ed KELLEy!NAMIOKA et al Linear Topological Spaces 37 MONK MathematicalLogic 38 GRAUERTIFRITZSCHE Several Complex Variables 39 ARVESON An Invitation to C*-Algebras 40 KEMENY/SNEUJKNAPP Denumerable Markov Chains 2nd ed 41 APOSTOL Modular Functions and Dirichlet Series in Number Theory 2nd ed 42 SERRE Linear Representations ofFinite Groups 43 GILLMAN/JERISON Rings ofContinuous Functions 44 KENDIG E1ementary Algebraic Geometry 45 LOEVE Probability Theory I 4th ed 46 LOEVE Probability Theory II 4th ed 47 MOISE Geometrie Topology in Dimensions and 48 SACHSlWu General Relativity for Mathematicians 49 GRUENBERGIWEIR Linear Geometry 2nded 50 EDWARDs Fermat's Last Theorem 51 KLINGENBERG A Course in Differential Geometry 52 HARTSHORNE Algebraic Geometry 53 MANIN A Course in Mathematical Logic 54 GRAvERIWATKINS Combinatorics with Emphasis on the Theory of Graphs 55 BROWNIPEARCY Introduction to Operator Theory I: Elements ofFunctional Analysis 56 MASSEY Algebraic Topology: An Introduction 57 CROWELLlFox Introduction to Knot Theory 58 KOBLITZ.p -adic Numbers,p-adic Analysis, and Zeta-Functions 2nd ed 59 LANG Cyclotomic Fields 60 ARNOLD MathematicalMethods in Classical Mechanics 2nd ed 61 WHITEIIEAD Elements ofHomotopy 62 KARGAPOLOvIMERLZJAKOV Fundamentals ofthe Theory ofGroups 63 BOLLOBAS Graph Theory 64 EDWARDS Fourier Series Vol I 2nd ed 65 WELLS Differential Analysis on Complex Manifolds 2nd ed 66 WATERHOUSE Introduction to Affine Group Schemes 67 SERRE Local Fields 68 WEIDMANN Linear Operators in Hilber! Spaces 69 LANG Cyclotomic Fields II 70 MAsSEY Singular Homology Theory 71 FARKAslKRA Riemann Surfaces 2nd ed (continued after index) Alain M Robert A Course in p-adic Analysis With 27 Figures Springer Alain M Robert Institut de Mathernatiques Universire de Neuchätel Rue Emile-Argand 11 Neuchätel CH-2oo7 Switzerland Editorial Board S Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA F.W Gehring Mathematics Department East Hall University of Michigan Ann ArOOr, MI 48109 USA K.A Ribet Mathematics Department University of California at Berkeley Berke1ey, CA 94720-3840 USA Mathematics Subject Classification (2000) : 11-01, IIE95, IISxx Library of Congress Cataloging-in-Publication Data Robert, Alain A course in p-adic analysis 1Alain M Robert p cm - (Graduate texts in mathematics ; 198) Includes bibliographical references and index ISBN 978-1-4419-3150-4 ISBN 978-1-4757-3254-2 (eBook) DOI 10.1007/978-1-4757-3254-2 I p-adic analysis I Title QA241, R597 2000 512'.74 - dc2I 11 Series 99-044784 Printed on acid-free paper © 2000 Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc in 2000 Softcover reprint of the hardcover 1st edition 2000 All rights reserved This work may not be translated or copied in whole or in part without the written permission ofthe publisher Springer Science+Bus iness Media, LLC, except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use of general descriptive names, trade names, trademarks , etc , in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone Production managed by Timothy Taylor; manufacturing supervised by Erica Bresler Typeset by TechBooks, Fairfax, VA 98765 321 Preface Kurt Hensel (1861-1941) discovered or invented the p-adic numbers' around the end of the nineteenth century In spite of their being already one hundred years old, these numbers are still today enveloped in an aura of mystery within the scientific community Although they have penetrated several mathematical fields, number theory, algebraic geometry, algebraic topology, analysis, , they have yet to reveal their full potential in physics, for example Several books on p-adic analysis have recently appeared: F Q Gouvea: p-adic Numbers (elementary approach); A Escassut: Analytic Elements in p-adic Analysis, (research level) (see the references at the end of the book), and we hope that this course will contribute to clearing away the remaining suspicion surrounding them This book is a self-contained presentation of basic p-adic analysis with some arithmetical applications * * * Our guide is the analogy with classical analysis In spite of what one may think, these analogies indeed abound Even if striking differences immediately appear between the real field and the p-adic fields, a better understanding reveals strong common features We try to stress these sirnilarities and insist on calculus with the p-adics, letting the mean value theorem play an important role An obvious reason for links between real/complex analysis and p-adic analysis is the existence of IThe letter p stands for a fixed prime (chosen in the list 2, 3, 5, 7,11, ) except when explicitly stated otherwise vi Preface an absolute value in both contexts.? But if the absolute value is Archimedean in reallcomplex analysis, if x =f: 0, for any y there is an integer n such that Inxl > lyl, it is non-Archimedean in the second context, narnely, it satisfies Inxl = I;X +x -+: +x,1 ::: Ix\ n tenns In particular, Inl ::: for all integers n , This implies that for any r ::: the subset of elements satisfying lxi::: r is an additive subgroup, even a subring if r =1 For such an absolute value, there is (except in a trivial case) exactly one prime p such that Ipl < 1.3 Intuitively, this absolute value plays the role of an order of magnitude If x has magnitude greater than 1, one cannot reach it from by taking a finite number ofunit steps (one cannot walk or drive to another galaxy!) Furthermore, Ipl < implies that Ipn I ~ 0, and the p-adic theory provides a link between characteristic and characteristic p The absolute value makes it possible to study the convergence oiformal power series, thus providing another unifying concept for analysis This explains the important role played by formal power series They appear early and thereafter repeatedly in this book, and knowing from experience the feelings that they inspire in our students, I try to approach them cautiously, as if to tarne them * * * Here is a short summary of the contents Chapter I: Construction of the basic p-adic sets Zp, Qp and Sp , Chapters II and III: Algebra, construction of C p and Qp, Chapters IV, V, and VI: Function theory , Chapter VII: Arithmetic applications I have tried to keep these four parts relatively independent and indicate by an asterisk in the table of contents the sections that may be skipped in a first reading I assume that the readers, (advanced) graduate students, theoretical physicists, and mathematicians, are familiar with calculus, point set topology (especially metric spaces, normed spaces), and algebra (linear algebra, ring and field theory) The first five chapters of the book are based solelyon these topics The first part can be used for an introductory course: Several definitions of the basic sets of p-adic numbers are given The reader can choose a favorite approach! Generalities on topological algebra are also grouped there 2Both Newton 's method for the determination of real roots of f = and Hensel's lemma in the p-adic context are applications of the existence of fixed points for contracting maps in a complete metric space 3Since the prime p is uniquely determined, this absolute value is also denoted by 1.lp However, since we use it systematically, and hardly ever consider the Archimedean absolute value, we simply write 1.1 Preface vii The second- more algebraic - part starts with a basic discussion of ultrametric spaces (Section 11.1) and ends (Section lIlA) with a discussion of fundamental inequalities and roots of unity (not needed before the study of the logarithm in Section VA) In between, the main objective is the construction of a complete and algebraically closed field C p , which plays a role similar to the complex field C of classical analysis The reader who is willing to take for granted that the p-adic absolute value has a unique extension I.IK to every finite algebraic extension K of Qp can skip the rest of Chapter II: If K and K ' are two such extensions, the restrictions of I.IK and I.IK' to K n K' agree This proves that there is a unique extension of the p-adic absolute value of Qp to the algebraic closure Q'; of Qp Moreover, if a E Aut (K /Qp), then x t-+ IxO" IK is an absolute value extending the p-adic one, hence this absolute value coincides with I.IK' This shows that o is isometrie If one is willing to believe that the completion = C p is also algebra ieally closed, most of Chapter III may be skipped as weIl In the third part , functions of a p-adic variable are examined In Chapter IV, continuous functions (and, in particular, locally constant ones) / : Zp -+ C p are systematically studied, and the theory culminates in van Hamme's generalization of Mahler's theory Many results conceming functions of a p -adic variable are extended from similar results conceming polynomials For this reason, the algebra of polynomials plays a central role, and we treat the systems of polynomials - umbral calculus - in a systematic way Then differentiability is approached (Chapter V): Strict differentiability plays the main role This chapter owes much to the presentation by W.H Schikhof: Ultrametrie Caleulus, an Introduetion to p-adic Analysis In Chapter VI, a previous acquaintance with complex analysis is desirable, since the purpose is to give the p-adic analogues of the classical theorems linked to the names ofWeierstrass, Liouville, Picard, Hadamard, Mittag-Leffler, among others In the last part (Chapter VII), some familiarity with the classical gamma function will enable the reader to perceive the similarities between the classical and the P: adic contexts Here, a means of unifying many arithmetic congruences in a general theory is supplied For example, the Wilson congruence is both generalized and embedded in analytical properties of the p-adic gamma function and in integrality properties of the Artin-Hasse power series I explain several applications of p -adic analysis to arithmetic congruences Qi * * * Let me now indicate one point that deserves more justifieation The study of metric spaces has developed around the classical examples of subsets of Rn (we make pictures on a sheet of paper or on the blackboard, both models of R ) , and a famous treatise in differential geometry even starts with "The nieest example 0/ ametrie spaee is Euclidean n-space Rn." This point of view is so widely shared that one may be 100to think that ultrametrie spaces are not genuine metric spaces! Thus the commonly used notation for metric spaces has grown on the paradigmatic model of subsets of Euclidean spaces For example, the "closed ball" of radius rand center a - defined by d(x, a) :s: r - is often denoted by B(a ;r) or Br(a) This notation comforts the belief that it is the closure of the "open ball" having the same viii Preface radius and center If the specialists have no trouble with the usual terrninology and notation (and may defend it on historical grounds), our students lose no opportunity to insist on its rnisleading meaning In an ultrametric space all balls of positive radius (whether defined by d(x, a) ::::: r or by d(x, a) < r) are both open and closed They are dopen sets Also note that in an ultrametric space, any point of a ball is a center of this ball The systematic appearance of totally disconnected spaces in the context of fractals also calls for a renewed view of metric spaces I propose using a more suggestive notation, B0 xpi ~ pi E! (Dwork) q e,,(x-x ) (q ld ppq = pi) 427 Basic Principles of Ultrametric Analysis in an Abelian Group (1) The strongest wins [x] » lyl = } Ix + yl = IxI (2) Equilibrium: All triangles are isoseeles (or equilateral ) a +b+c = 0, (3) Competitivity al [c] « Ibl + a2 + + an = there is i#- j such that =} lai = Ibl =} lad = lajl = max jakl (4) A dream realized (an)n:::O is a Cauchy sequence ~ dta.; an+l) ~ o (5) Another dream come true (in a complete group) Ln:::o an converges ~ an ~ o When Ln:::oan converges, Ln:::o lan I may diverge, but I Lanl n:::O s sup lanl = max lanl and the infinite version of (3) is valid (6) Stationarity of the absolute value an ~ a #- = } there is N with lanl = lal/or n 2: N Conventions, Notation, Terminology We use the abbreviations , iff " if and only if," := "equal by definition," == nontriv ial equality • is the "e nd of proof" (or "absence of proof") sign In a statement: (I), (ii), always denote equivalent prop erties In the table of contents, an asterisk * before a section indicates that it will not be used later and may be ornitted in a first reading Set Theory P(E) power set of E : Set of subsets of E ; : Empty set A C B means "x E A ==> x E B" hence : AcE {:=:} A E P(E) (certain authors denote this inclusion by ~) When AC B CE , B - A = B \ Adenotes the complement of A in B, E - A = ACis the complement of a subset ACE A subset of E having only one element is a singleton set: x E E ==> {x} E P(E) U: Disjoint union symbol, partition of a set EI : Set of families (or functions) I ~ E E(l) : Set of familie s I ~ E having components equal to the base point of E (the neutral element in a group G, the in a ring A ) except f or finitely many indices Let f : E ~ F , x t-+ fex) be a map Then fis injective when x =1= y ==> fex) =1= f( y) , namely f is one-to-one, or equivalently when fex) = f(y) ==> x = y, fis surj ective when f(E) = F (namely f is onto), f is bijective when it is one-to-one and onto 432 Conventions, Notation, Tenninology The characteristic function of a subset ACE is the function gl(x) I = glA(X) = { if XE A, if x!f A Fundamental Sets of Numbers N = {O, 1,2, , n , } C Z c Q c R c C, N* = {I, 2, , n, } = N>o When pE {2, 3, 5, 7,11 , } is a prime, F p = ZlpZ p I n means p divides n, ptn means p does not divide n, pV 11 n means that p" is the highest power of p dividing n, R>o = {x ER : x > O}, R:;:o = {x ER: x ~ O} , [a, b) : interval a ::::: x < b Z(p) = {alb: a E Z, b ~ 1, b prime to p} C Q , Z[I/p]={ap v:aEZ, VEZ}CQ When a > and S C R, a S = las : SES} C R >o, e.g., pZ C pQ C R>o [x] E Z integral part of x ER: [x] ::::: x < [x] + (x) fractional part ofx ER: x = [x] + (x) gcd: Greatest common divisor ; lern: Least common multiple äij : Kronecker symbol (= ifi = i, = otherwise) Groups, Rings and Modules A x : Multiplicative group of units (i.e., invertible elements) in a ring A A[X]: Polynomial ring in one indetenninate X and coefficients in the ring A, a monic polynomial fis a polynomial having leading coefficient 1: X" + an_IX n- + + ao if deg f = n A[[X]]: Formal power series ring A{X}: Restricted power series over a valued ring A (Chapter V: Power series with coefficients - 0) A[X] C A{X} C A[[X]] An integral domain is a commutative ring A =1= {O} having no zero divisor K = Frac A : Fraction field of an integral domain A In particular, K(X) = Frac A[X] : Rational fractions, K((X» = Frac A[[X]] (:J K(X» : Formal Laurent series ring A [1/ q ]: Partial fraction ring corresponding to denominators in {l, q , q2, .}, where q is not a zero divisor in the ring A If G is an abelian group , then {g E G : gn = e for some integer n ~ I} is the torsion subgroup of G: In particular, Jl(A) denotes the group of roots of unity in a commutative ring A, X Jl = Jl(C ) = Jlpoo x Jl(p) , where Jlp oo : pth-power roots ofunity (p-Sylow subgroup of Jl), Jl(p) : Roots of unity having order prime to p, Jln(A) = {x E A : x n = I}: nth roots of unity in the ring A A pair ofhomomorphisms A ~ B ~ Cis exact when f(A) = ker g A short exact sequence (SES) is an exact pair with f injective and g surjective ; hence Cis a quotient of B by f(A) ~ A, written _ A ~ B ~ C - for additive groups (replace by for multiplicative groups) Conventions , Notation, Tenninology 433 Fields, Extensions Characteristic of a field K : Either or the prime p such that p I K = E K, in which case the prime field F p is contained in K For each prime p, the group F; is cyclic; when the prime p is odd, the squares in F; make up a subgroup of index two , kernel of the Legendre symbol (~) = ± = In a field (or a ring) of characteristic p we have (x + y)P x P + y" , KU : Aigebraic closure of a field K ; when K = KU is algebraically closed of characteristic 0, J-Ln(K) is cyclic and isomorphie to Z/nZ pi (K) = K U {oo} denotes the projective line over the field K Topology, Metric Spaces The closure of a subset A C X (X being a topological space) is denoted by A A Hausdorffspace is a topological space X in which for every pair of distinct points, it is possible to find disjoint neighborhoods ofthese points: Equivalently, the diagonal ßx is closed in the product X x X The diameter of a sub set A C X with respect to ametrie d is diam(A) = 8(X) = SUPx,yEA d(x, y) ~ 00 We say that A is bounded when diam(A) < 00 The distance ofa point x E X to a subset AC Xis d(x, A) = infuEA d(x, a), d(x , A) = XE A The balls in ametrie space (X, d) are denoted by B:::r(a) = B :::r(a; X) = {x EX : d(x , a) ~ r} : closed (dressed) ball, B